×

Polygon shortening makes (most) quadrilaterals circular. (English) Zbl 1012.53055

The Gage-Grayson-Hamilton theorem states that if a closed curve is deformed by means of vectorial velocities equal to the curvature vectors it tends to a circular shape. This paper contains a discretization of this statement, replacing the curves by closed polygons.
Let \(m\) and \(r\) denote the center and radius of the circumcircle of three points \(x, y, z\) in the plane and let \(c(x,y,z)\) be the vector \(c(x,y,z) := (m-y)/r^2\) (here \(c(x,y,z)\) is the zero vector iff \(x,y,z\) are collinear). Commonly, a family \(\{y_i(t)\mid i \in{\mathbb Z}/ n{\mathbb Z}\), \(t \in I = [0, T)\}\) of \(n\)-gons is said to be evolving by its Menger curvature if \(dy_i/dt = c(y_{i-1}(t), y_i(t), y_{i+1}(t))\). This means that at any moment each vertex \(y_i(t)\) of the polygon moves towards the center of the instantaneous circumcircle of \(y_{i-1}(t), y_i(t), y_{i+1}(t)\) with a scalar velocity inverse to the radius of this circle.
The following results are worked out in the paper:
If all \(y_i(t_0)\) lie on a common straight line for \(t_0 \in I\) then \(y_i(t) = y_i(t_0)\) for all \(t\) in \(I\).
If all \(y_i(t_0)\) lie on a common circle \(k\) for \(t_0 \in I\) then the family \(y_i(t)\) is generated by a pure dilatation having its center in the center of \(k\).
The larger part of the paper deals with quadrilaterals (\(n=4\)). There the main result reads as follows:
If \(x_0, x_1, x_2, x_3\) is an arbitrary start-quadrilateral then there exist an interval \(I := [0, T)\) and a unique family \(\{y_0(t), y_1(t), y_2(t), y_3(t)\), \(t \in I\}\) of quadrilaterals evolving by its Menger curvature such that \(y_i(0) = x_i\). Moreover for \(t\rightarrow T\) the quadrilateral either tends to a cocyclical shape (all four points lying on a common circle) or (in exceptional cases) to a collinear quadruple or a parallelogram.

MSC:

53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
53A04 Curves in Euclidean and related spaces
PDFBibTeX XMLCite
Full Text: DOI